Properties

Label 2160.4.a.bh
Level $2160$
Weight $4$
Character orbit 2160.a
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1765.1
Defining polynomial: \( x^{3} - x^{2} - 11x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 5 q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + \beta_1 q^{7} + (2 \beta_{2} - \beta_1 - 9) q^{11} + ( - \beta_{2} + 2 \beta_1 - 1) q^{13} + (5 \beta_{2} - 3 \beta_1 + 5) q^{17} + ( - 3 \beta_{2} - 3 \beta_1 + 26) q^{19} + ( - 6 \beta_{2} + \beta_1 - 35) q^{23} + 25 q^{25} + ( - 5 \beta_{2} + \beta_1 + 39) q^{29} + ( - 8 \beta_{2} - \beta_1 + 69) q^{31} - 5 \beta_1 q^{35} + ( - 19 \beta_{2} + 19 \beta_1 - 40) q^{37} + ( - 3 \beta_{2} - 16 \beta_1 + 100) q^{41} + (10 \beta_{2} + 3 \beta_1 + 161) q^{43} + (19 \beta_{2} - 17 \beta_1 - 101) q^{47} + (12 \beta_{2} - 13 \beta_1 - 5) q^{49} + ( - 11 \beta_{2} + 8 \beta_1 - 164) q^{53} + ( - 10 \beta_{2} + 5 \beta_1 + 45) q^{55} + (3 \beta_{2} - 30 \beta_1 - 80) q^{59} + (20 \beta_{2} - 37 \beta_1 - 148) q^{61} + (5 \beta_{2} - 10 \beta_1 + 5) q^{65} + (36 \beta_{2} + \beta_1 + 174) q^{67} + ( - 17 \beta_{2} + 52 \beta_1 + 56) q^{71} + (27 \beta_{2} - 11 \beta_1 - 292) q^{73} + (14 \beta_{2} + 4 \beta_1 - 50) q^{77} + (3 \beta_{2} + 10 \beta_1 + 701) q^{79} + ( - 53 \beta_{2} + 26 \beta_1 + 14) q^{83} + ( - 25 \beta_{2} + 15 \beta_1 - 25) q^{85} + (13 \beta_{2} + 30 \beta_1 - 756) q^{89} + (11 \beta_{2} - 27 \beta_1 + 532) q^{91} + (15 \beta_{2} + 15 \beta_1 - 130) q^{95} + (29 \beta_{2} - 3 \beta_1 - 464) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} - 27 q^{11} - 3 q^{13} + 15 q^{17} + 78 q^{19} - 105 q^{23} + 75 q^{25} + 117 q^{29} + 207 q^{31} - 120 q^{37} + 300 q^{41} + 483 q^{43} - 303 q^{47} - 15 q^{49} - 492 q^{53} + 135 q^{55} - 240 q^{59} - 444 q^{61} + 15 q^{65} + 522 q^{67} + 168 q^{71} - 876 q^{73} - 150 q^{77} + 2103 q^{79} + 42 q^{83} - 75 q^{85} - 2268 q^{89} + 1596 q^{91} - 390 q^{95} - 1392 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 11x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -3\nu^{2} + 3\nu + 22 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -6\nu^{2} - 6\nu + 48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + 2\beta _1 + 4 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} - 2\beta _1 + 92 ) / 12 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.48079
2.89055
1.59024
0 0 0 −5.00000 0 −24.7900 0 0 0
1.2 0 0 0 −5.00000 0 5.60585 0 0 0
1.3 0 0 0 −5.00000 0 19.1841 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.4.a.bh 3
3.b odd 2 1 2160.4.a.bp 3
4.b odd 2 1 1080.4.a.f 3
12.b even 2 1 1080.4.a.l yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.a.f 3 4.b odd 2 1
1080.4.a.l yes 3 12.b even 2 1
2160.4.a.bh 3 1.a even 1 1 trivial
2160.4.a.bp 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2160))\):

\( T_{7}^{3} - 507T_{7} + 2666 \) Copy content Toggle raw display
\( T_{11}^{3} + 27T_{11}^{2} - 1272T_{11} + 8044 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 507T + 2666 \) Copy content Toggle raw display
$11$ \( T^{3} + 27 T^{2} - 1272 T + 8044 \) Copy content Toggle raw display
$13$ \( T^{3} + 3 T^{2} - 1629 T + 19553 \) Copy content Toggle raw display
$17$ \( T^{3} - 15 T^{2} - 9708 T + 420688 \) Copy content Toggle raw display
$19$ \( T^{3} - 78 T^{2} - 10635 T + 766504 \) Copy content Toggle raw display
$23$ \( T^{3} + 105 T^{2} - 11088 T - 502268 \) Copy content Toggle raw display
$29$ \( T^{3} - 117 T^{2} - 5484 T + 275060 \) Copy content Toggle raw display
$31$ \( T^{3} - 207 T^{2} - 19632 T + 3717376 \) Copy content Toggle raw display
$37$ \( T^{3} + 120 T^{2} + \cdots - 22578642 \) Copy content Toggle raw display
$41$ \( T^{3} - 300 T^{2} - 124740 T + 9666000 \) Copy content Toggle raw display
$43$ \( T^{3} - 483 T^{2} + 13440 T + 370864 \) Copy content Toggle raw display
$47$ \( T^{3} + 303 T^{2} - 145332 T + 2148016 \) Copy content Toggle raw display
$53$ \( T^{3} + 492 T^{2} + 29628 T - 8122896 \) Copy content Toggle raw display
$59$ \( T^{3} + 240 T^{2} + \cdots - 117125056 \) Copy content Toggle raw display
$61$ \( T^{3} + 444 T^{2} + \cdots - 202420030 \) Copy content Toggle raw display
$67$ \( T^{3} - 522 T^{2} - 531759 T + 6471308 \) Copy content Toggle raw display
$71$ \( T^{3} - 168 T^{2} + \cdots + 521430912 \) Copy content Toggle raw display
$73$ \( T^{3} + 876 T^{2} + \cdots - 13506642 \) Copy content Toggle raw display
$79$ \( T^{3} - 2103 T^{2} + \cdots - 297054405 \) Copy content Toggle raw display
$83$ \( T^{3} - 42 T^{2} + \cdots - 368484664 \) Copy content Toggle raw display
$89$ \( T^{3} + 2268 T^{2} + \cdots - 159259392 \) Copy content Toggle raw display
$97$ \( T^{3} + 1392 T^{2} + \cdots - 80387830 \) Copy content Toggle raw display
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